Asymptotics of the eigenvalues of a boundary value problem for the operator Schrödinger equation with boundary conditions nonlinearly dependent on the spectral parameter

On the space $H_{1}=L_{2}(H, [ 0, 1 ] )$, where $H$ is a separable Hilbert space, we study the asymptotic behavior of the eigenvalues of a boundary value problem for the operator Schrödinger equation for the case when one, and the same, spectral parameter participates linearly in the equation and quadratically in the boundary condition. Asymptotic formulae are obtained for the eigenvalues of the considered boundary value problem.